Generally speaking, a good chunk of the number systems mathematicians worked with involve taking some existing number system (tracing back to the natural numbers), finding some operation that takes you outside that system, and then building a new system to cover that hole. A system that has no such holes given a particular operation is said to "close over" that operation. So, integers close over subtraction; rationals close over division, and complex numbers close over exponentiation. Since all of these systems ultimately derived from operations over the natural numbers, it makes sense to also call them numbers, even though they might lack certain symmetries or properties of simpler number systems.
Also, the word "measure" is already taken by a different concept.
"Natural number", "real number", "complex number" are.
The fact that they all contain the string "number" is completely irrelevant.
"A number is a mathematical object used to count, measure, and label."
Meanwhile natural numbers like "two" most certainly do exist as a quantitative attribute of a set.
Show me how you can count with reals.
Rational and real numbers represent the most intuitive concept of quantity with different cardinality; natural numbers are more basic in theory but a restricted special case in most application (they can only count "countable" objects and they aren't dense); complex numbers compensate unnatural weirdness with important applications; then there are other very number-like niches (e.g. p-adic numbers, quaternions, algebraic extension fields...) and number-based arbitrary constructions (multidimensional vector spaces, matrices and tensors...).
> Show me how you can count with reals.
Like you count with natural numbers, plus many other possibilities because there are more numbers to count with. The same applies to rational numbers (which exist more than real numbers), complex numbers, etc.
Wait, did you say there are more rationals than reals? Isn't that the other way around?
I don't know if that's a slip of the tongue or I'm missing something in my recall of basic math lessons
Real numbers are of course more numerous than rational numbers, just like unicorn horns are more numerous than horse horns, but it's an entirely different question.
Is it because more people understand rationals than reals? Ok, that would make rationals more existent, as they exist in more brains.
For example, if you start with number '1' and apply operator _successor_ (or "adding one more"), you get the *natural numbers*, which are a mapping from the size of sets to strings of operators {'1', 'successor(1)', 'successor(successor(1))', ... }. You can use this process to define an order: number A is smaller than B if you can repeatedly apply _successor_ to A and generate B in a finite time. (Not the best definition, I know, but bear with me for a second).
If you reverse the _successor_ operator, you get the _predecessor_, which can be used to dismount large numbers and make them smaller. If you apply _predecessor_ to '1', you get 'predecessor(1)' which doesn't match any of the *natural* numbers as defined in the paragraph above. However, this new number is useful because you can map it to a collection without elements, allowing you to count a set of size 'zero'. This set doesn't exist, but it has a well defined size thanks to the process explained above. (You may as well apply _predecessor_ to 'zero' and count *negative numbers*, which allow you to create a mapping with debt, and thus count _I owe you_ amounts that don't exist in the physical world either).
Now as for real and complex numbers, they can't count physical objects as you said because at some point you can't measure smaller and smaller magnitudes (though you can create orders with them, even if it doesn't directly involve the _successor_ operator). But mathematicians use them to count the size of _infinite sets_, which being immaterial never lose precision; you just know you can repeat their defining processes once and again, showing that there exist a mapping between any step in the process and the instance of the real or complex number that correspond to its size.